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Properties

Label 2-70-35.12-c1-0-0
Degree $2$
Conductor $70$
Sign $0.927 - 0.373i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.0749 + 0.279i)3-s + (0.866 − 0.499i)4-s + (2.20 − 0.378i)5-s − 0.289i·6-s + (0.126 + 2.64i)7-s + (−0.707 + 0.707i)8-s + (2.52 + 1.45i)9-s + (−2.03 + 0.935i)10-s + (−2.81 − 4.87i)11-s + (0.0749 + 0.279i)12-s + (−1.42 − 1.42i)13-s + (−0.806 − 2.51i)14-s + (−0.0593 + 0.645i)15-s + (0.500 − 0.866i)16-s + (−5.12 − 1.37i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.0432 + 0.161i)3-s + (0.433 − 0.249i)4-s + (0.985 − 0.169i)5-s − 0.118i·6-s + (0.0477 + 0.998i)7-s + (−0.249 + 0.249i)8-s + (0.841 + 0.486i)9-s + (−0.642 + 0.295i)10-s + (−0.848 − 1.46i)11-s + (0.0216 + 0.0807i)12-s + (−0.396 − 0.396i)13-s + (−0.215 − 0.673i)14-s + (−0.0153 + 0.166i)15-s + (0.125 − 0.216i)16-s + (−1.24 − 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749983 + 0.145341i\)
\(L(\frac12)\) \(\approx\) \(0.749983 + 0.145341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-2.20 + 0.378i)T \)
7 \( 1 + (-0.126 - 2.64i)T \)
good3 \( 1 + (0.0749 - 0.279i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.81 + 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.42 + 1.42i)T + 13iT^{2} \)
17 \( 1 + (5.12 + 1.37i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.94 - 3.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.290 + 1.08i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.15iT - 29T^{2} \)
31 \( 1 + (3.33 - 1.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.86 + 1.30i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.21iT - 41T^{2} \)
43 \( 1 + (-1.85 + 1.85i)T - 43iT^{2} \)
47 \( 1 + (1.52 + 5.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.33 - 0.357i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.73 - 4.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 + 2.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.218 + 0.816i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (-1.45 + 5.42i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.41 - 3.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.67 - 5.67i)T + 83iT^{2} \)
89 \( 1 + (5.96 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.63 - 6.63i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.03069789881621589569830772183, −13.63042959747958888750704832307, −12.72214001808167408513283281430, −11.11920427675390542523581665800, −10.18168507749680496442110351101, −9.066030974980261201395000014415, −8.046431387517954042476418666583, −6.28711416929015966766111266866, −5.21275965683295963987468921767, −2.37046207890397897291589424561, 2.03768544672533025692614727603, 4.53043958240600486729400809543, 6.67935135699068995937512245708, 7.40220021994503878612552538465, 9.254570735152117448214152338250, 10.07674203200132231646674290396, 10.94282235670674311587353388422, 12.66507191768093066052650969418, 13.28358530152748847508894511634, 14.74407078192859634935083839242

Graph of the $Z$-function along the critical line